On the projective geometry of rational homogeneous varieties

Abstract

Abstract.We determine the varieties of linear spaces on rational homogeneous varieties, provide explicit geometric models for these spaces, and establish basic facts about the local differential geometry of rational homogeneous varieties. Mathematics Subject Classification (2000). 14M15, 20G05.

Quantum Cohomology of Minuscule Homogeneous Spaces

“…The Fano variety of lines through a given point can be identified with the spinor variety G Q (5, 10) in its minimal embedding (the projectivization of a half-spin representation). In particular its degree equals 12 (see [LM,3.1]). Applying proposition 3.1, we get the relation…”

“…It was proved in [LM,Theorem 4.8] that when P is defined by a long simple root, F o is homogeneous under of G/P by suppressing the marked node and marking the nodes that were connected to it. For example, if G/P = E 7 /P 7 then F o is a copy of the Cayley plane E 6 /P 6 ≃ E 6 /P 1 , whose dimension is 16 -hence c 1 (E 7 /P 7 ) = 18.…”

Deformation Rigidity of Odd Lagrangian Grassmannians

“…Although this is identical to Lemma 3.6 of [2], we give details of the proof for the convenience of the reader since the proof there is omitted. For the case of (even) symplectic Grassmannians, the following lemma was originally proven in Proposition 3.2.1 of [9] or Corollary 5.5 of [13].…”

On the third secant variety

“…When X = G/P with G simple, this is [11,Theorem 4.3] and [5]. (This is true for any minimally embedded homogeneous variety G/P I , with G simple, where I indexes the deleted simple roots, as long as I does not contain an "exposed short root" in the language of [11].…”